Periods and entropy for Lorenz-like maps
Annales de l'Institut Fourier, Volume 39 (1989) no. 4, p. 929-952

We characterize the set of periods and its structure for the Lorenz-like maps depending on the rotation interval. Also, for these maps we give the best lower bound of the topological entropy as a function of the rotation interval.

Nous caractérisons l’ensemble des périodes et la structure des applications de type Lorenz en fonction de l’intervalle de rotation. Pour ces applications nous donnons la meilleure borne inférieure de l’entropie topologique comme une fonction de l’intervalle de rotation

@article{AIF_1989__39_4_929_0,
     author = {Alsed\`a, Lluis and Llibre, J. and Misiurewicz, M. and Tresser, C.},
     title = {Periods and entropy for Lorenz-like maps},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {39},
     number = {4},
     year = {1989},
     pages = {929-952},
     doi = {10.5802/aif.1195},
     zbl = {0678.34047},
     mrnumber = {91e:58146},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1989__39_4_929_0}
}
Alsedà, Lluis; Llibre, J.; Misiurewicz, M.; Tresser, C. Periods and entropy for Lorenz-like maps. Annales de l'Institut Fourier, Volume 39 (1989) no. 4, pp. 929-952. doi : 10.5802/aif.1195. http://www.numdam.org/item/AIF_1989__39_4_929_0/

[ALMM] L. Alsedà, J. Llibre, F. Mañosas and M. Misiurewicz, Lower bounds for the topological entropy of continuous maps of the circle of degree one, Nonlinearity, 1 (1988), 463-479. | MR 89m:58119 | Zbl 0663.54023

[ALMS] L. Alsedà, J. Llibre, F. Misiurewicz and C. Simó, Twist periodic orbits and topological entropy for continuous maps of the circle of degree one which have a fixed point, Ergod. Th. and Dynam. Sys., 5 (1985), 501-518. | MR 87h:58185 | Zbl 0592.54037

[ALM] L. Alsedà, J. Llibre and M. Misiurewicz, Periodic orbits of maps of Y, Trans. Amer. Math. Soc., 313 (1989), 475-538. | MR 90c:58145 | Zbl 0803.54032

[BGMY] L. Block, J. Guckenheimer, M. Misiurewicz and L. S. Young, Periodic points and topological entropy of one-dimensional maps, Springer, Lect. Notes in Math., 819 (1980), 18-39. | MR 82j:58097 | Zbl 0447.58028

[CGT] A. Chenciner, J. M. Gambaudo and C. Tresser, Une remarque sur la structure des endomorphismes de degré 1 du cercle, C.R. Acad. Sc. Paris, 299, Sér. I (1984), 145-148. | MR 86b:58102 | Zbl 0584.58004

[GPTT] J. M. Gambaudo, I. Procaccia, S. Thomae and C. Tresser, New universal scenarios for the onset of chaos in Lorenz type flows, Phys. Rev. Lett., 57 (1986), 925-928.

[GT] J. M. Gambaudo and C. Tresser, Dynamique régulière ou chaotique. Applications du cercle ou de l'intervalle ayant une discontinuité, C.R. Acad. Sc., Paris, 300, Ser. I (1985), 311-313. | MR 86i:58085 | Zbl 0595.58031

[G1] J. Guckenheimer, A strange, strange attractor, in the Hopf Bifurcation and its applications, Eds. J. E. Marsden and M. McCracken, Appl. Math. Sc. (Springer), 19 (1976), 368-381.

[G2] J. Guckenheimer, Bifurcations of Dynamical Systems in : Dynamical Systems C.I.M.E. Lectures, Progress in Mathematics, 8, Birkhäuser, Boston, 1980. | MR 82g:58065 | Zbl 0451.58025

[H1] F. Hofbauer, The maximal measure for linear mod. one transformations, J. London Math. Soc., 23 (1981), 92-112. | MR 82c:28040 | Zbl 0431.54025

[H2] F. Hofbauer, Periodic points for piecewise monotonic transformation, Ergod. Th. and Dynam. Sys., 5 (1985), 237-256. | MR 87i:58141 | Zbl 0572.54036

[L] E. N. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci., 20 (1963), 130-141.

[M1] M. Misiurewicz, Periodic points of maps of degree one of a circle, Ergod Th. and Dynam. Sys., 2 (1982), 221-227. | MR 84j:58101 | Zbl 0508.58038

[M2] M. Misiurewicz, Twist sets for maps of a circle, Ergod. Th. and Dynam. Sys., 4 (1984), 391-404. | MR 86m:58135 | Zbl 0573.58019

[M3] M. Misiurewicz, Rotation intervals for a class of maps of the real line into itself, Ergod. Th. and Dynam. Sys., 6 (1986), 117-132. | MR 87k:58131 | Zbl 0615.54030

[MS] M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63. | MR 82a:58030 | Zbl 0445.54007

[MT] J. Milnor and W. Thurston, On iterated maps of the interval, in Dynamical Systems, Ed. J. C. Alexander, Lecture Notes in Math., 1342 (1988), 465-563. | MR 90a:58083 | Zbl 0664.58015

[P] W. Parry, The Lorenz attractor and a related population model, Lect. Notes in Math., Springer, 729 (1979), 169-187. | MR 81f:58027 | Zbl 0431.92022

[R] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., 8 (1957), 477-493. | Zbl 0079.08901

[RT] F. Rhodes and C. L. Thompson, Rotation numbers for monotone functions on the circle, J. London Math. Soc., 34 (1986), 360-368. | MR 88b:58127 | Zbl 0623.58008